X-MACE: Transferable Excited-State Modeling

• X-MACE is a computational framework that delivers high-fidelity excited-state potential energy surfaces and smooth handling of conical intersections using neural network and quantum Monte Carlo methods.
• It employs message-passing atomic cluster expansion and DeepSets to ensure robust chemical transferability and scalable simulations across diverse molecular geometries.
• The approach achieves dramatic efficiency gains and precise modeling of photochemical phenomena, enabling accurate nonadiabatic dynamics in large and complex environments.

Transferable Excited-State Modeling (X-MACE) encompasses a set of computational frameworks that enable high-fidelity, data-efficient, and transferable simulations of excited-state potential energy surfaces (PESs) and their intersections in molecular systems. X-MACE refers to both a family of neural network architectures—rooted in message-passing atomic cluster expansion (MACE) and its variants—as well as transferable excited-state quantum Monte Carlo (QMC) methods that collectively address the challenges of excited-state modeling: smooth interpolation across conical intersections, chemical transferability, scalability to large environments, and efficient reuse of ground-state knowledge (Barrett et al., 18 Feb 2025, Schätzle et al., 25 Mar 2025, Barrett et al., 28 Feb 2025, Axelrod et al., 2021).

1. Theoretical Foundations and Core Models

Transferable excited-state modeling with X-MACE is grounded in the Born–Oppenheimer approximation, where the molecular Hamiltonian at a fixed nuclear geometry $R$ admits stationary electronic eigenstates: $\hat{H}(R)\,\Psi^i(r|R) = E^i(R)\,\Psi^i(r|R)$ with

$$\hat{H}(R) = -\frac{1}{2}\sum_{i}\nabla_i^2 - \sum_{i,I}\frac{Z_I}{|r_i-R_I|} + \sum_{i<j}\frac{1}{|r_i-r_j|}$$

The archetypal quantum Monte Carlo instantiation employs a parameterized Slater–Jastrow form: $$\Psi_\theta(r|R) = e^{J_\theta(r,R)}\sum_{k=1}^K \det[A_\theta^{(k)}(r,R)]$$ The loss functional at a given geometry incorporates the Rayleigh quotient, spin penalties, and orthogonality constraints to hierarchically optimize $N_s$ target states: $$\mathcal{L}_R(\Psi_{\theta,1}, \ldots, \Psi_{\theta,N_s}) = \sum_{i=1}^{N_s}\Big[ E_\theta^i(R) + \beta\,\langle S^2 \rangle_\theta^i(R) + \sum_{j < i} \alpha_{ij} |\langle \Psi_{\theta,i}(R)|\Psi_{\theta,j}(R)\rangle|^2 \Big]$$ Geometric and electronic state transferability is realized by sharing parameters $\theta$ across all sampled $R\in\mathcal{R}$, while dynamic state ordering ensures adiabatic surface continuity through conical intersections (Schätzle et al., 25 Mar 2025).

X-MACE as machine-learned PES architecture extends the MACE formalism. Each molecule is represented as a graph $G=(V,E)$ of atom positions and atomic numbers. Local atomic environments are encoded by equivariant message passing, culminating in atom-wise features $h_i^{(T)}$, which were classically mapped to adiabatic state energies. X-MACE, however, deploys DeepSets to obtain permutation-invariant latent descriptors $I$, which are decoded back to energies through a learned Hermitian "companion" matrix. This ensures smoothness even through conical intersections (Barrett et al., 18 Feb 2025).

2. Treatment of Conical Intersections and Nonadiabatic Couplings

Conical intersections, regions where two or more adiabatic PESs become degenerate and exhibit cusp-like non-smoothness, are crucial for simulating nonradiative photochemical transitions. Standard electronic structure and ML PES models struggle near these topological features. In X-MACE, the DeepSets latent invariant ensures the final decoded eigenvalues ($\hat{E}_i$) are smooth functions of geometry—even as adiabatic state labels permute. The reconstruction loss enforces

$$L_{\mathrm{recon}} = \frac{1}{n}\sum_{i=1}^n (\hat{E}_i - E_i)^2$$

By fitting not to adiabatic states but to permutation-invariant summarizations, cusps are replaced by smooth manifolds in the latent space (Barrett et al., 18 Feb 2025).

For quantum Monte Carlo approaches, state continuity through intersections is maintained by dynamic ordering and overlap terms. The interstate overlap,

$$O^i_{mn} = \langle \Psi_{\theta,i}(R_m) | \Psi_{\theta,i}(R_n) \rangle$$

remains near unity on an adiabatic sheet, allowing for precise detection of true crossings.

Nonadiabatic couplings, essential for surface hopping and other nonadiabatic molecular dynamics, are treated in X-MACE using a smoothed observable: $$S_{mn} = C_{mn}\cdot (E_n - E_m), \qquad C_{mn} = \langle \psi_m | \nabla_R \psi_n \rangle \approx \frac{\langle \psi_m | \nabla_R H | \psi_n \rangle}{E_n - E_m}$$ This enables robust training and evaluation around degenerate points without singularities (Barrett et al., 18 Feb 2025).

3. Transferability Strategies

A core innovation of X-MACE is transferability across (i) geometry, (ii) state, and (iii) chemical space. This is accomplished through:

• Geometry: Neural network architectures receive nuclear coordinates $R$ and encode them via nuclear encoders and self-attention modules. Thus, a single set of parameters $\theta$ models the entire relevant domain of $R$ (Schätzle et al., 25 Mar 2025).
• State: Extensive parameter sharing across $N_s$ electronic state nets, followed by synchronized averaging after each training step, facilitates efficient joint learning of correlated PESs.
• Chemical space: Transfer learning leverages foundational ground-state potentials ("MACE-OFF") as the initialization for excited-state PES training. Only the final DeepSets readout is reinitialized, and all earlier parameters (message-passing kernels, radial basis functions, and equivariant features) are preserved, yielding strong performance even with minimal excited-state data (Barrett et al., 18 Feb 2025, Barrett et al., 28 Feb 2025).

In the context of long-range interactions and QM/MM environments, FieldMACE incorporates the multipole expansion (up to $\ell_{\max}=3$) via an equivariant, attention-weighted message-passing scheme, enabling linear scaling with system size and explicit solvent effects (Barrett et al., 28 Feb 2025).

4. Computational Efficiency and Benchmark Performance

X-MACE methods are characterized by dramatic reductions in wall-clock and sample complexity relative to traditional single-point approaches. Empirical benchmarks include:

System	Metric/Reference	Cost Reduction	Energy MAE
Ethylene S₀/S₁ torsion	MAE_rel (meV)	~5× fewer iterations	29.4(1) (torsion), 4.7(7) (pyramidal) (Schätzle et al., 25 Mar 2025)
Carbon dimer (C₂), 8 states	MAE_rel (meV)	~3× faster	69(1) vs. SHCI (Schätzle et al., 25 Mar 2025)
CH₂NH₂⁺ (3-state PES grid)	MAE_rel (meV)	~100×	96(2) vs. MR-CISD (Schätzle et al., 25 Mar 2025)
Butene (ML)	MAE (eV)	—	0.0151 (X-MACE+AE) (Barrett et al., 18 Feb 2025)
Furan in water (FieldMACE)	Population curves	Transfer: ~30 points	Reproduces QM, versus failure from scratch (Barrett et al., 28 Feb 2025)

Fine-tuning from ground-state MACE-OFF models with only 1–10% excited-state data yields lower MAEs (e.g., energies < 0.01 eV) than training de novo; when >30% data is available, converged accuracy is equal (Barrett et al., 18 Feb 2025). Linear scaling ($O(N)$ in atom number for NN PES evaluation) enables large-scale explicit-environment excited-state dynamics.

5. Case Applications: Photochemistry and Large-System Dynamics

X-MACE has demonstrated robust performance across a diverse suite of photochemical benchmarks:

• Ethylene S₀/S₁ torsion: Correct conical intersection at $\phi \approx 98.5^\circ$ resolved, with $\Delta E_{ij}(R)$ matching high-level references.
• Carbon dimer dissociation: Simultaneous optimization of eight lowest states over bond lengths, all crossings and adiabatic gaps are captured.
• Methylenimmonium cation ($CH_2NH_2^+$): 2D PES with conical intersections at $\tau=0^\circ$; surface features and relative gaps resolved over the grid (Schätzle et al., 25 Mar 2025).
• Butene/ethene/propene/fulvene: ML X-MACE with DeepSets improves energy-gap accuracy near conical intersections by 45% relative to conventional MACE.
• Chromophore dataset: Transferable ML potentials exhibiting 30% lower energy MAE and 15% lower force MAE on held-out molecules (Barrett et al., 18 Feb 2025).
• Furan in water: FieldMACE transfer models reproduce excited-state surface-hopping population curves using only 30 QM points (Barrett et al., 28 Feb 2025).

Diabatic neural network approaches ("DANN") extended X-MACE principles to the explicit modeling and screening of 3,100 photoswitchable azobenzene derivatives, achieving $\sim 10^5$–$10^6\times$ acceleration over quantum chemistry (Axelrod et al., 2021).

6. Implications, Generalization, and Outlook

Transferable excited-state modeling with X-MACE enables routine ab-initio-level simulation of photochemical phenomena, including on-the-fly nonadiabatic dynamics without external fitting. Critical implications include:

• Generation of training data for excited-state force fields with high-fidelity reference labeling.
• Direct computation of nonadiabatic coupling elements via overlaps or smooth NAC predictors, facilitating surface hopping and Ehrenfest dynamics.
• Scalability to large systems (peptides, protein chromophores, nanomaterials) due to approximately linear cost in system size and number of geometries.
• Chemical-space extrapolation and transfer: fine-tuning from ground-state foundational models results in improved data efficiency and generalization to unseen chemical motifs (Barrett et al., 18 Feb 2025, Barrett et al., 28 Feb 2025).
• Explicit environmental and solvent modeling via long-range multipole expansions (FieldMACE) (Barrett et al., 28 Feb 2025).

A plausible implication is the use of X-MACE-derived PESs and couplings as drop-in modules for ML-enhanced surface-hopping packages, enabling controlled studies of photodynamics in previously inaccessible chemical regimes. These advances position X-MACE as a principal tool for high-accuracy, data-efficient, and transferable excited-state simulations in modern theoretical chemistry.